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Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, we will learn about arithmetic series and how they work. Learn also how to find the pattern of an arithmetic sequence along with finding the sum.

Arithmetic Sequence

Arithmetic sequences, or strings of numbers where each number is the previous number plus a constant, can be found all around us. Simply start counting and you are saying the numbers of a very famous arithmetic sequence, our numbers! It goes 1, 2, 3 and so on. Notice how each number is the previous number plus one? This difference between our numbers is called the common difference. We write an arithmetic sequence with curly brackets and commas between the numbers. So our counting numbers can be written {1, 2, 3, . . .} with the three dots in the end telling us that the sequence continues indefinitely.

Arithmetic Series

If we add up a few or all of the numbers in our sequence, then we have what is called an arithmetic series. In other words, an arithmetic series is the sum of the numbers in our arithmetic sequence. You will come across math problems that will ask you to find the sum of an arithmetic sequence as you continue on in your math lessons. Keep watching, and I will show you a formula you can use to find this sum. But before we can do that, we need to find the common difference.

Finding the Common Difference

Finding our common difference is an easy process. What you do is you take any pair of successive numbers and you subtract the first from the second. Now, you take another pair and subtract the first from the second to see if it also has the same difference. If you get the same difference then you have found your common difference. Each arithmetic sequence will have its own common difference.

For example, the sequence {4, 6, 8, 10, . . .} is an arithmetic sequence because it has a common difference of two, because each pair of successive numbers has a difference of two between them. We have 6 - 4 = 2. We also have 8 - 6 = 2. Ten minus eight is also two.

Look at the sequence {2, 5, 8, 11, . . .}. Does this sequence have a common difference? We subtract 5 - 2, we get 3. What is 8 minus 5? Is it also 3? Yes. Okay; so far, so good. Finally, what is 11 - 8? Oh good, it is also 3. So our common difference is 3.

Arithmetic Series Sum

Now that we know how to find our common difference, we can go about and find our arithmetic series sum. We can use the formula n / 2 (2a (n - 1)d).

The n represents the number of terms in our series that we are adding up. If we want to add the first 10 numbers, for example, then n = 10. Our problem will tell us how many numbers we are adding up. The a represents the beginning term of our sequence, and d is the common difference. So for our sequence {4, 6, 8, 10, . . .} from before, our a is 4, and our d is 2. If we want to find the sum of the first five terms, then n = 5. Let's go ahead and calculate this. We plug in all our values and evaluate to see what we get.

We get an answer of 40.

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Let's try finding the first ten terms of the sequence {5, 10, 15, 20, . . .}. First, before we use the formula, we want to see if this is actually an arithmetic sequence. To do this, we look to see if we have a common difference. If we don't, then we don't have an arithmetic sequence, and we can stop right there. We subtract 10 - 5. We get 5. Now, what is 15 - 10 and 20 - 15? Are they also equal to 5? If they are, then this is an arithmetic sequence, and we can go ahead with our formula. If they are not, then we stop and say that this sequence is not an arithmetic sequence and, therefore, has no arithmetic series. In this case, this is indeed an arithmetic sequence. The common difference here is 5. Now we can use our formula to find our arithmetic series for the first ten terms, for n - 10. Our a is 5 since that is our first number. Our d is 5 since that is our common difference. We plug these numbers into our formula and evaluate.

Our answer is 275, and we are done.

Lesson Summary

Let's review what we've learned now. We've learned that arithmetic sequences are strings of numbers where each number is the previous number plus a constant. The common difference is the difference between the numbers. If we add up a few or all of the numbers in our sequence, then we have what is called an arithmetic series. To find the sum of the arithmetic series, we first need to find our common difference.

Whenever we come across such a problem, our first objective is to see if our sequence actually has a common difference and is actually an arithmetic sequence. To find the common difference, we check to see if each pair of successive numbers gives us the same answer when we subtract the first from the second. If they do, then this difference is our common difference. Once we've found our common difference, we label that with d, our first number of our sequence with a and the number of terms we are summing up with n. We then use the formula n / 2 (2a (n - 1)d) to get our answer.

Learning Outcomes

By the end of this lesson you should be able to:

Define arithmetic sequence and series

Identify the common difference of an arithmetic sequence

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